So they basically operate like the Peacekeepers in Farscape?
Originally a Peacekeeping force, they effectively maintain the rule of the establishment by conducting invasions, lightning raids and punitive attacks, to keep civilized planets in line. Other than that, planets maintain their own governments (nominally). A bit like the Jem'Hadar, although I would say, better done.
Species like the Dosi were seemingly able to continue their cultural practices relatively unmolested, so long as order, the rule of law (as the Founders saw it), and stability were maintained. Memory Alpha claims the Dominion was envisioned as an anti-Federation:
Robert Hewitt Wolfe: "We just felt it was time to give a face to the Gamma Quadrant. Voyager was going to be wandering through the Delta Quadrant from place to place, meeting new people every week, and we wanted to make the Gamma Quadrant distinctly different from that, by creating the Dominion, a sort of unifying anti-Federation in a way, just to give it a completely different character. Instead of like the big mysterious out there, which all the other Star Trek shows had done, and Voyager was going to do, it was a very specific, dangerous, nasty Other, so that part of the motivation."
The Dominion was first mentioned in "Rules of Acquisition", then in "Sanctuary" and next in "Shadowplay", before finally being encountered in "The Jem'Hadar". The initial idea was to make the Dominion a kind of anti-Federation, similar in structure but with very different ideologies. The Dominion was to represent a wide array of alien races, just as does the Federation (as opposed to the mono-species Klingon Empire, Romulan Star Empire, and Cardassian Union), but it was to be fascist-like, ruled by coercion and domination, in contrast to the cooperation and freedom of the Federation. As Robert Hewitt Wolfe explains, "The Dominion was not monolithic, it wasn't just the Romulans or the Cardassians, they were distinct in that they were the Dominion, they were, like the Federation, a collection of different races. But unlike the Federation, they were bound together by fear and extortion, whereas the Federation is bound together by noble thoughts and love and friendship and all that good stuff. So in a lot of ways, they were the mirror image of the Federation."
Initially, the plan was was for numerous different species to be seen on Dominion vessels and involved in various parts of the Dominion's activities, although eventually, only three "main" species were firmly established: the Founders, the Jem'Hadar, and the Vorta, although the Karemma were also a member.
Now that could have been really interesting; if Dominion ships were manned by all sorts of multi-ethnic crew instead of just Jem'Hadar. I think they lost a little bit of that 'anti-Federation' idea along the way, and they became a bit of a bland empire, from a nice concept.
Also, although they call the Romulans and Klingons mono-cultures, in reality, I'm sure their empires contain a great many species - the Romulan Empire, being like Rome, probably has similar diversity, and the Klingon Empire was originally was an 'anti-Federation' itself - but of course, they only allowed ethnic Romulans or Klingons in their militaries.